Problem

Source: 2022 Saudi Arabia November Camp Test 2.4 BMO + EGMO TST

Tags: algebra, functional equation, functional



Find all functions $f : R \to R$ such that $$2f(x)f(x + y) -f(x^2) =\frac{x}{2}(f(2x) + 4f(f(y)))$$for all $x, y \in R$.